Designation E2041 − 13´1 Standard Test Method for Estimating Kinetic Parameters by Differential Scanning Calorimeter Using the Borchardt and Daniels Method1 This standard is issued under the fixed des[.]
Trang 1Designation: E2041−13
Standard Test Method for
Estimating Kinetic Parameters by Differential Scanning
This standard is issued under the fixed designation E2041; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision A number in parentheses indicates the year of last reapproval A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε 1 NOTE—Warning statements were editorially corrected throughout in September 2013.
1 Scope
1.1 This test method describes the determination of the
kinetic parameters of activation energy, Arrhenius
pre-exponential factor, and reaction order using the Borchardt and
Daniels2 treatment of data obtained by differential scanning
calorimetry This test method is applicable to the temperature
range from 170 to 870 K (−100 to 600°C)
1.2 This treatment is applicable only to smooth exothermic
reactions with no shoulders, discontinuous changes, or shifts in
baseline It is applicable only to reactions with reaction order
n ≤ 2 It is not applicable to acceleratory reactions and,
therefore, is not applicable to the determination of kinetic
parameters for most thermoset curing reactions or to
crystalli-zation reactions
1.3 The values stated in SI units are to be regarded as
standard No other units of measurement are included in this
standard
1.4 This test method is similar, but not equivalent to,
ISO 11357, Part 5, that contains provisions for additional
information not supplied by this test method
1.5 This standard does not purport to address all of the
safety concerns, if any, associated with its use It is the
responsibility of the user of this standard to establish
appro-priate safety and health practices and determine the
applica-bility of regulatory limitations prior to use.
2 Referenced Documents
2.1 ASTM Standards:3
E473Terminology Relating to Thermal Analysis and Rhe-ology
E537Test Method for The Thermal Stability of Chemicals
by Differential Scanning Calorimetry E698Test Method for Arrhenius Kinetic Constants for Thermally Unstable Materials Using Differential Scan-ning Calorimetry and the Flynn/Wall/Ozawa Method E967Test Method for Temperature Calibration of Differen-tial Scanning Calorimeters and DifferenDifferen-tial Thermal Ana-lyzers
E968Practice for Heat Flow Calibration of Differential Scanning Calorimeters
E1142Terminology Relating to Thermophysical Properties E1445Terminology Relating to Hazard Potential of Chemi-cals
E1641Test Method for Decomposition Kinetics by Thermo-gravimetry Using the Ozawa/Flynn/Wall Method E1970Practice for Statistical Treatment of Thermoanalytical Data
2.2 ISO Standards:4
ISO 11357Part 5: Determination of Temperature and/or Time of Reaction and Reaction Kinetics
3 Terminology
3.1 Definitions—Specific technical terms used in this test
method are defined in TerminologiesE473,E1142, andE1445,
including calibration, calorimeter, differential scanning
calorimetry, enthalpy, peak, reaction, repeatability, reproducibility, and slope.
4 Summary of Test Method
4.1 A test specimen is heated at a linear rate in a differential scanning calorimeter or other suitable calorimeter through a region of exothermic reaction behavior The rate of heat evolution, developed by a chemical reaction, is proportional to the rate of reaction Integration of the heat flow as a function of time yields the total heat of a reaction
1 This test method is under the jurisdiction of ASTM Committee E37 on Thermal
Measurements and the direct responsibility of Subcommittee E37.01 on Calorimetry
and Mass Loss.
Current edition approved Sept 15, 2013 Published September 2013 Originally
approved in 1999 Last previous edition approved in 2008 as E2041 – 08 ε1 DOI:
10.1520/E2041-13E01.
2Borchardt, H.J., Daniels, F., Journal of the American Chemical Society, Vol 79,
1957, pp 41–46.
3 For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org For Annual Book of ASTM
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
4 Available from American National Standards Institute (ANSI), 25 W 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959 United States
Trang 24.2 The Borchardt and Daniels2 data treatment is used to
derive the kinetic parameters of activation energy, Arrhenius
pre-exponential factor, and reaction order from the heat flow
and total heat of reaction information obtained in 4.1 (see
Section5)
5 Basis of Methodology
5.1 Kinetic reactions may be modeled with a number of
suitable equations The Borchardt and Daniels2method makes
use of the rate equation to describe the dependence of the rate
of reaction on the amount of material present
where:
dα/dt = reaction rate (min−1)
α = fraction reacted (dimensionless),
k(T) = rate constant at temperature T (min−1), and
n = reaction order (dimensionless)
5.2 For a reaction conducted at temperature (T), the rate
equation ofEq 1, may be cast in its logarithmic form:
ln@dα/dt#5 ln@k~T!#1nln@1 2 α# (2)
This equation has the form of a straight line, y = mx + b,
where a plot of the logarithm of the reaction rate (ln[dα/dt])
versus the logarithm of the fraction remaining ln[1 − α] yields
a straight line, the slope of which is equal to n and the intercept
is equal to ln[k(T)].
5.3 The Borchardt and Daniels2model also makes use of the
Arrhenius equation to describe how the reaction rate changes
as a function of temperature:
k~T!5 Z e ·E/RT (3) where:
Z = Arrhenius pre-exponential factor (min−1),
E = Activation energy (J mol−1),
T = Absolute temperature (K), and
R = Gas constant (= 8.314 J mol−1K−1)
5.4 The Arrhenius equation Eq 3 also may be cast in its
logarithmic form:
ln@k~T!#5 ln@Z#2 E/RT (4)
The equation has the form of a straight line, y = mx + b,
(where y ≡ ln[k(T)], m ≡ E/R, x ≡1/T and b ≡ ln[Z]) where a plot
of the logarithm of the reaction rate constant (ln[k(T)]) versus
the reciprocal of absolute temperature (l/T) produces a straight
line, the slope of which is equal to −E/R and the intercept of
which is ln[Z].
5.5 As an alternate toEq 2 and 4, the rate and Arrhenius
equations may be combined and cast in its logarithmic form:
ln@dα/dt#5 ln@Z#1nln@1 2 α#2 E/RT (5)
The resultant equation has the form z = a + bx + cy (where
z ≡ ln[dα/dt], ln[Z] ≡ a, b ≡ n, x ≡ ln[1 − α], c ≡ E/R, and y ≡
l/T) and may be solved using multiple linear regression data
treatment
5.6 The values for dα/dt, (1 − α) and T needed to solveEq
2,Eq 4andEq 5, are experimental parameters obtained from
a single linear heating rate DSC experiment scanning through the temperature region of the reaction exotherm as shown in
Fig 1 5.7 Kinetic results obtained by this test method may be compared with those obtained by Test MethodE698
6 Significance and Use
6.1 This test method is useful in research and development 6.2 The determination of the appropriate model for a chemi-cal reaction or transformation and the values associated with its kinetic parameters may be used in the estimation of reaction performance at temperatures or time conditions not easily tested This use, however, is not described in this test method
7 Interferences
7.1 Because of its simplicity and ease of use, the Borchardt and Daniels2 method is often the method of choice for characterization of the kinetic parameters of a reaction system The Borchardt and Daniels method, like all tools used to evaluate kinetic parameters, is not applicable to all cases The user of this test method is expressly advised to use this test method and its results with caution
7.2 Tabulated below are some guidelines for the use of the Borchardt and Daniels2method
7.2.1 The approach is applicable only to exothermic reac-tions
N OTE 1—Endothermic reactions are controlled by the kinetics of the heat transfer of the apparatus and not by the kinetics of the reaction. 7.2.2 The reaction under investigation must have a constant mechanism throughout the whole reaction process In practice, this means that the reaction exotherm upon heating must be smooth, well shaped (as inFig 1) with no shoulders, multiple peaks or discontinuous steps
7.2.3 The reaction must be nth order Confirmation of an nth
order reaction may be made by an isothermal experiment such
as that described inAppendix X1
7.2.4 Typical reactions which are not nth order and to which
Borchardt and Daniels2kinetic may not be applied for predic-tive purposes include many thermoset curing reactions and crystallization transformations
7.2.5 The nth order kinetic reactions anticipate that the value of n will be small, non-zero integers, such as 1 or 2 Values of n greater than 2 or that are not simple fractions, such
as 1⁄2 = 0.5, are highly unlikely and shall be viewed with caution
7.2.6 The Borchardt and Daniels2method assumes tempera-ture equilibrium throughout the whole test specimen This means that low heating rates, (that is, <10 K/min), small specimen sizes (<5 mg) and highly conductive sealed specimen containers, for example, aluminum, gold, platinum, etc., should
be used
7.3 Since milligram quantities of specimen are used, it is essential that the specimen be homogeneous and representative
of the test sample from which they are taken
7.4 Toxic or corrosive effluents, or both, may be released when heating the test specimen and may be harmful to
E2041 − 13´
Trang 3personnel or to the apparatus Operating with a venting or
exhaust system is recommended
8 Apparatus
8.1 Differential Scanning Calorimeter (DSC)—The
instru-mentation required to provide the minimum differential
scan-ning calorimetric capability for this method includes the
following:
8.1.1 DSC Test Chamber, composed of the following:
8.1.1.1 Furnace(s), to provide uniform controlled heating of
a specimen and reference to a constant temperature at a
constant rate within the applicable temperature range of this
test method
8.1.1.2 Temperature Sensor, to provide an indication of the
specimen/furnace temperature to 60.01 K
8.1.1.3 Differential Sensor, to detect heat flow difference
between the specimen and reference equivalent to 1 µW
8.1.1.4 A means of sustaining a test chamber environment
of purge gas at a rate of 10 to 50 mL/min
N OTE 2—Typically, 99.9+ % pure nitrogen, helium, or argon is
employed Use of dry purge gas is recommended and is essential for
operation at subambient temperatures.
8.1.2 Temperature Controller, capable of executing a
spe-cific temperature program by operating the furnace(s) between
selected temperature limits, that is, 170 to 870 K, at a rate of
temperature change of up to 10 K/min constant to 60.1 K/min
8.1.3 Data Collection Device, to provide a means of
acquiring, storing, and displaying measured or calculated signals, or both The minimum output signals required for DSC are heat flow, temperature, and time
8.2 Containers (pans, crucibles, vials, etc.), that are inert to
the specimen and reference materials, and which are of suitable structural shape and integrity to contain the specimen and reference in accordance with the specific requirements of this test method
8.3 While not required, the user will find useful calculator or computer and data analysis software to perform the necessary least squares best fit or multiple linear regression data treat-ments required by this test method
8.4 Balance—to weigh specimens, or containers, or both, to
610 µg with a capacity of at least 100 mg
9 Calibration
9.1 Perform any calibration procedures recommended by the apparatus manufacturer in the instrument operator’s manual
9.2 Calibrate the DSC temperature signal over the range of the reaction using Test MethodE967
9.3 Calibrate the DSC heat flow signal using PracticeE968
FIG 1 Idealized DSC Curve
Trang 410 Procedure
10.1 Weigh 1 to 10 mg of test specimen to a precision of
610 µg into a sample container and hermetically seal the
container Weigh the specimen and container to 610 µg Load
the test specimen into the apparatus using an equivalent empty
specimen container as the reference Close the DSC sample
chamber and prepare the apparatus for an experimental run
N OTE 3—This test method is based upon a “non-self heating”
assump-tion Combinations of specimen size and reaction kinetics that produce
heat flow greater than 8 mW fail this assumption and produce erroneous
results Small specimen sizes may be used to obtain this critical non-self
heating assumption.
10.2 Equilibrate the specimen at a temperature 40 K below
the first exothermic behavior
N OTE 4—This temperature may be determined from a previously
recorded exploratory run using Test Method E537
10.3 Heat the test specimen at a rate of 5 K/min to a
temperature 10 K higher than the completion of the exothermic
reaction as indicated by the return to baseline Record the heat
flow and sample temperature throughout this region
N OTE 5—Other heating rates (<10 K/min) may be used but shall be
indicated in the report Agreement of results undertaken at several heating
rates will provide confidence in the method and efficacy of the results.
10.4 Cool the specimen container to ambient temperature
and reweigh Record and report any change in mass from that
observed in10.1prior to the test
10.5 Calculate reaction order (n), activation energy (E), and
Arrhenius pre-exponential factor (Z) according to the
proce-dures in Section11
11 Calculation
11.1 Construct a linear baseline from a point on the baseline
before the reaction exotherm to a point on the baseline after the
reaction
11.2 Construct a perpendicular line from the baseline to the
peak of the thermal curve and record this value in mW Only
results for which the maximum heat flow (as expressed by this
line) are less than 8 mW shall be used in these calculations If
the heat flow at the peak maximum is greater than 8 mW,
reduce the specimen size or heating rate and rerun the
experiment (see Note 3)
11.3 Integrate the total peak area bounded by the peak itself
and the constructed baseline to obtain the heat of the reaction
(∆H) in mJ.
11.4 Identify the temperatures that correspond
approxi-mately to 10 and 90 % of the peak area obtained in11.3
11.5 Select a temperature interval which provides a
mini-mum of ten equally-spaced values between the temperature
limits determined in11.4
11.6 At each of the ten temperatures identified in 11.5,
record the rate of reaction (dH/dt ) in mW, temperature (T) in
K and heat of reaction remaining (∆H T) in mJ as illustrated in
Fig 1
N OTE 6—It is convenient to prepare a table of these values.
11.7 For each of the fractional areas obtained in 11.6, determine the fraction remaining (1 − α) and the fractional rate
of reaction (dα/dt) using the following equation:
~1 2 α!5 ∆H T /∆H (6)
N OTE 7—In this and all subsequent calculations, retain all available significant figures rounding only the final result to the number of significant figures described in Section 13
N OTE 8—The values for (1 − α) should range between 0.9 and 0.1 depending upon the values selected in 11.4 and 11.5
11.8 Calculate the reciprocal of absolute temperature for each value determined in11.6and11.7(seeNote 7)
N OTE 9—Often, it is convenient to report the value of reciprocal temperature in units of kK −1
11.9 Calculate the natural logarithm of the rate of reaction
(ln[dα/dt ]) for each of the values determined in11.6and11.7
(see Note 7)
11.10 Determine the values for n, s n , E, s E, ln[Z], and sln[Z]
by either Method A or Method B below
11.11 Method A:
11.11.1 Assume a value for n = 1.0.
11.11.2 Calculate the value for n ln[1 − α] for each value
determined in11.6and11.7(seeNote 7)
11.11.3 Calculate the value for ln[k(T)] using:
ln@k~T!#5 ln@dα/dt#2 nln@1 2 α# (8) for each value determined in11.6and11.7(seeNote 7)
11.11.4 Prepare a plot of ln[k(T)] versus l/T such as that in
Fig 2
N OTE 10—The uncertainty introduced into results by linearizing
non-linear data are discussed in Journal of Chemical Education.5
11.11.4.1 This plot should result in a straight line If it does
not, assume a new value for n and repeat the steps in 11.11.2 – 11.11.4.1until a straight line is obtained Report the value of
n, which yields a straight line A curve that is concave upward
indicates that the value of n is too large while one that is concave downward indicates that n is too small.
11.11.5 Using a multiple linear regression technique (see PracticeE1970), determine a slope (m), intercept (b), standard deviation in slope (s m ), and standard deviation in intercept (s b)
for this straight line The slopes m and s m have the units of
1/kK Intercepts b and s bare dimensionless
11.11.6 Calculate the value for activation energy and
stan-dard deviation in activation energy (s E) using the following equations:
where:
R = 8.314 J mol-1K-1
11.11.7 Determine the value of ln[Z] and standard deviation
in ln[Z] (that is, s ln Z ) from the values of b and s bin11.11.5
5 Zielinski, T J., and Allendoerfer, R D., “Least squares fitting of nonlinear data
in the undergraduate laboratory,” Journal of Chemical Education, Vol 74, 1997, pp.
1001–1007.
E2041 − 13´
Trang 5ln@Z#5 b (11)
11.11.8 Calculate the logarithm of the reaction rate constant
at any temperature of interest using the values from 11.11.6,
11.11.7, andEq 14
11.12 Method B:
11.12.1 Alternatively, the values for n, s n , E, s E , ln Z and
sln Zmay be determined simultaneously using a multiple linear
regression data treatment applied to the following equation:
ln@dα/dt#5 ln@Z#5 nln@1 2 α#2 E/RT. (13)
11.13 Calculate the logarithm of the reaction rate constant at
any temperature of interest using the values from11.12.1and
the equation:
ln@k@~T!##5 ln@Z#2 E/RT (14)
12 Report
12.1 The report shall include the following information:
12.1.1 Complete identification and description of the
mate-rial tested, including source, manufacturing codes, etc
12.1.2 Description of the instrument and software used for
the test
12.1.3 Experimental conditions including test specimen
mass, heating rate, temperature range of test, specimen
container, purge gas type, and flow rate
12.1.4 Description of the data treatment used including
name and version of any software packages and whether
Method A or Method B is used
12.1.5 The values and their standard deviations for reaction
order (n 6 s n ) activation energy (E 6 sE) in kJ/mol and
logarithm of the Arrhenius frequency factor (ln[Z] 6 sln Z),
with Z in 1/min Report all values to one position to the right
of decimal point
12.1.6 Report any values of logarithm of the rate constant at temperature of interest
12.1.7 The original thermal curve
12.1.8 The dated version of this test method use
13 Precision and Bias
13.1 An interlaboratory test was conducted in 2002 to determine the precision and bias of Method B of E2041-01 using trityl azide (for example, azidotriphenlymethane) The results from a minimum of 12 laboratories, using 5 replicates each (that is, 44 degrees of freedom) are used to provide the information6listed below Seven instrument models from three manufacturers were used and the data was evaluated using three different software programs
13.2 Precision:
13.2.1 Within laboratory variability may be described using the repeatability value (r) obtained by multiplying the standard deviation by 2.8 The repeatability value estimates the 95 % confidence limits That is, two results obtained in the same
6 Supporting data have been filed at ASTM International Headquarters and may
be obtained by requesting Research Report RR:E37-1028 Contact ASTM Customer Service at service@astm.org.
FIG 2 Plot of ln[k (T)] versus 1/T
Trang 6laboratory should be considered suspect (at the 95 %
confi-dence level) if they differ by more than the repeatability value
r
13.2.2 The within laboratory repeatability relative standard
deviation for activation energy (E), logarithm of the
pre-exponential factor (log [Z]) and reaction order (n) was found to
be 3.0, 3.4, and 6.6 %, respectively
13.2.3 Between laboratory variability may be described
using the reproducibility value (R) obtained by multiplying the
standard deviation by 2.8 The repeatability value estimates the
95 % confidence limits That is, two results obtained in
different laboratories, should be considered suspect (at the 95
% confidence level) if they differ by more than the
reproduc-ibility value R
13.2.4 The between laboratory reproducibility relative
stan-dard deviation for activation energy (E), logarithm of the preexponential factor (log [Z]), and reaction order (n) was
found to be 9.8, 9.8 and 22 %, respectively
13.3 Bias:
13.3.1 Bias is the difference between the value obtained and that of a reference material There are no known standard values for kinetic parameters for trityl azide and so bias may not be estimated (SeeX2.1.)
14 Keywords
14.1 activation energy; Arrhenius frequency factor; Bor-chardt and Daniels kinetics; differential scanning calorimetry (DSC); kinetics; pre-exponential factor; reactions
APPENDIXES
(Nonmandatory Information)
X1 TEST FOR nth ORDER OR AUTOCATALYTIC REACTIONS
X1.1 The Borchardt and Daniels2method is applicable only
to nth order reactions It is not applicable to accelerating
reactions This appendix describes a useful test procedure for
verifying a given reaction is nth order.
X1.2 Weigh 1 to 5 mg of the test specimen into a sample
container and hermetically seal the container Do not load the
test specimen into the apparatus Load an equivalent empty
specimen container as the reference Close the DSC sample
chamber, and prepare the apparatus for an experimental run
X1.3 Select the temperature corresponding to 10 % peak
area from10.3as the isothermal test temperature T Equilibrate
the apparatus at this test temperature
X1.4 Initiate the experiment recording heat flow as a
func-tion of time
X1.5 Open the DSC sample chamber and load the test
specimen into the apparatus Immediately close the sample
chamber Record the thermal curve for 20 min or until the exothermic event is complete, that is, the rate of heat flow
approaches zero (Warning—Burn hazard The sample
cham-ber and its heat shields and covers are hot presenting a burn hazard to the operator Exercise great care in this operation Use protective safety equipment to ensure the safety of the operator.)
X1.6 Observe the shape of the resultant thermal curve Heat-flow curves, which reach a maximum heat flow value within a few seconds after placement of the test specimen in
the apparatus and then decay away, are likely to be due to nth
order reactions Heat-flow curves, which begin low, build to a maximum (after tens of seconds) and then decay away, are likely to be due to autocatalyzed reactions SeeFig X1.1and
Fig X1.2 for example curves
E2041 − 13´
Trang 7FIG X1.1 Heat Flow Curve for an nth Order Reaction
Trang 8X2 COMPARATIVE RESULTS
X2.1 The trityl azide (used in the interlaboratory test
refer-enced in Section13) was used as a test specimen in additional
interlaboratory tests A comparison of mean values may be
useful to the user
X2.2 The mean values determined by DSC Method B of
E2041 – 01 for activation energy (E), logarithm of the
pre-exponential factor (log [Z]), and reaction order (n) were
found to be 165.1 kJ/mol, 17.2 (with Z expressed in min-1) and
1.32, respectively
X2.3 Trityl azide was also used in an interlaboratory test
(see Research Report RR:E27-10027) for DSC variable heating
rate Test Method E698 In that study, the mean values for
activation energy and logarithm of the pre-exponential factor
were 145.5 kJ/mol and 15.2 (with Z expressed in min-1),
respectively (with 8 degrees of freedom) In Test MethodE698,
the reaction order (n) is fixed at 1.00 These values are
statistically (student t-test) different from those obtained in this
test
X2.4 Trityl azide was also evaluated in an intralaboratory test using the TGA variable heating rate test standard Test Method E1641 In this study, the mean values for activation energy and logarithm of the pre-exponential factor were 73.0
kJ/mol and 7.13 (with Z expressed in min-1), respectively In Test Method E1641, the reaction order (n) is fixed at 1.00.
These values are statistically different (student t-test) from those obtained in this test or those of Test MethodE698 X2.5 Trityl azide was also evaluated in an intralaboratory test using modulated thermogravimetry an approach that is not yet an ASTM International standard8In this work, the mean values for activation energy and logarithm of the
pre-exponential factor were 87.5 kJ/mol and 8.87 (with Z expressed
in min-1), respectively In modulated thermogravimetry, the
reaction order (n) is fixed at 1.00 These values are statistically
different (student t-test) from those obtained in this test or those
of Test Method E698
7 Supporting data have been filed at ASTM International Headquarters and may
be obtained by requesting Research Report RR:E27-1002 Contact ASTM Customer
Service at service@astm.org.
8 Blaine, R.L., Hahn, B.K., “Obtaining Kinetic Parameters by Modulated
Thermogravimetry,” Journal of Thermal Analysis, Vol 54, 1998, pp 658–704.
FIG X1.2 Heat-Flow Curve for an Autocatalyzed Reaction
E2041 − 13´
Trang 9ASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentioned
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